Corollary 3.3.6.10. Let $X$ be a Kan complex. Then the comparison map $\rho _{X}^{\infty }: X \rightarrow \operatorname{Ex}^{\infty }(X)$ is a homotopy equivalence.
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Corollary 3.3.6.10. Let $X$ be a Kan complex. Then the comparison map $\rho _{X}^{\infty }: X \rightarrow \operatorname{Ex}^{\infty }(X)$ is a homotopy equivalence.
Proof. Since $\operatorname{Ex}^{\infty }(X)$ is also a Kan complex (Proposition 3.3.6.9), it will suffice to show that $\rho _{X}^{\infty }$ is a weak homotopy equivalence (Proposition 3.1.6.13), which follows from Proposition 3.3.6.7. $\square$